Novel Functional Representation of Self-similar functions

Abstract:

We propose a functional framework for the representation and construction of self-similar structures

based on recursive functional compositions, offering an alternative to purely geometric iterated function

systems. We introduce a functional notion of self-similarity for real-valued functions, illustrated through

log-periodic constructions, where complexity emerges under repeated scaling and composition. Unlike

classical constructions, which characterise self-similar sets through affine or nonlinear transformations

on geometric domains, the present work develops a formulation of self-similar generation in terms of

functional operators acting on functions. The proposed framework defines a class of recursively generated functions whose limiting behaviour exhibits self-similar characteristics. Conditions under which

these functions display irregular or jagged structures are discussed, including cases where smoothness is

preserved while local complexity persists. Numerical experiments illustrate the emergence of self-similarlike behaviour under functional iteration, and a root-finding experiment highlights sensitivity to initial

conditions consistent with chaotic dynamics. The results demonstrate that recursive functional constructions can serve as a viable analytical representation of self-similar structures, providing a foundation for

further theoretical investigation and quantitative analysis.